Non-universal threshold behaviour of random resistor networks with anomalous distributions of conductances

The exponent t which describes the conductivity of a random resistor network near the percolation threshold is generally independent of the form of the distribution h( sigma ) of the non-zero conductors. However, in cases where h approximately sigma ^{- alpha}, t comes to depend on alpha . Here this problem is discussed using a combination of the Skal-Shklovskii-de Gennes model, and renormalisation ideas, with the conclusion that t( alpha )=(d-2) nu +(1- alpha )^-1 when this is greater than t_un, with a continuous transition at a value of alpha -which may be greater than zero.